Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The automorphism groups of Kummer surfaces in characteristic two and their complex analogues

Shigeyuki Kondo, Shigeru Mukai

TL;DR

The paper develops a unified, lattice-theoretic approach to determining automorphism groups of Kummer surfaces that arise as $K3$-surfaces in characteristic two, focusing on Jacobians of genus-2 curves (with varying $p$-rank) and products of elliptic curves. By leveraging Borcherds–Conway reflection theory, the Leech lattice, and Mordell–Weil lattices of elliptic fibrations, the authors construct explicit involutions and organize them into concrete (often infinite) automorphism groups, typically as semidirect products with finite 2-groups. They also establish complex-analytic analogues for the same Picard lattices, present explicit quartic models with prescribed singularities, and show how the automorphism group structure depends on the underlying NS-lattice and the geometric configuration of $(-2)$-curves. The results unify characteristic-2 and complex cases, provide explicit generators, and connect automorphism groups to rich lattice-theoretic data (Göpel tetrads, Weber hexads, and Leech-root configurations).

Abstract

We calculate the automorphism group of the Kummer surface associated with a curve of genus 2 or the product of two elliptic curves in characteristic two under the assumption that the Kummer surface is a $K3$ surface. Moreover we discuss the complex $K3$ surfaces with the same Picard lattice as these Kummer surfaces. The paper has two appendices.

The automorphism groups of Kummer surfaces in characteristic two and their complex analogues

TL;DR

The paper develops a unified, lattice-theoretic approach to determining automorphism groups of Kummer surfaces that arise as -surfaces in characteristic two, focusing on Jacobians of genus-2 curves (with varying -rank) and products of elliptic curves. By leveraging Borcherds–Conway reflection theory, the Leech lattice, and Mordell–Weil lattices of elliptic fibrations, the authors construct explicit involutions and organize them into concrete (often infinite) automorphism groups, typically as semidirect products with finite 2-groups. They also establish complex-analytic analogues for the same Picard lattices, present explicit quartic models with prescribed singularities, and show how the automorphism group structure depends on the underlying NS-lattice and the geometric configuration of -curves. The results unify characteristic-2 and complex cases, provide explicit generators, and connect automorphism groups to rich lattice-theoretic data (Göpel tetrads, Weber hexads, and Leech-root configurations).

Abstract

We calculate the automorphism group of the Kummer surface associated with a curve of genus 2 or the product of two elliptic curves in characteristic two under the assumption that the Kummer surface is a surface. Moreover we discuss the complex surfaces with the same Picard lattice as these Kummer surfaces. The paper has two appendices.

Paper Structure

This paper contains 30 sections, 59 theorems, 101 equations, 23 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kummer surfaces associated with curves $C$ of genus 2 in characteristic 2 whose Jacobian have $p$-rank 2
  4. Kummer surfaces associated with curves $C_1$ of genus 2 whose Jacobian have $p$-rank 1
  5. The Leech lattice and the reflection group of a hyperbolic lattice
  6. Leech roots in terms of a fixed oval
  7. The odd unimodular hyperbolic lattices and their even parts
  8. Mordell--Weil lattices
  9. The automorphism group of the Kummer surface associated with a curve of genus 2 whose Jacobian has $p$-rank 2
  10. The Kummer surface associated with a curve whose Jacobian has $p$-rank 1
  11. Kummer surfaces associated with the product of two ordinary elliptic curves
  12. The case $E$ is not isogenous to $F$.
  13. The case $E=F$.
  14. Kummer surfaces associated with the product of an ordinary elliptic curve and a supersingular elliptic curve
  15. Quartics with $4D_4$ singularities: complex version of Theorem \ref{['MainCurve']}
  16. ...and 15 more sections

Key Result

Proposition 2.1

(Laszlo--Pauly LPLP2, Katsura--Kondō KK2; See Appendix B also.) (i) The Kummer quartic surface $S$ associated with $C$ is given by the equation (ii) The quartic surface $S$ has exactly four rational double points of type $D_4$ and contains four tropes $T_1, T_2, T_3, T_4$ defined by $x=0, y=0, z=0, w=0$ respectively.

Figures (23)

  • Figure 1: MOG as a projective plane (CS)
  • Figure 2: Root lattice $R_0$ of type $D_4$
  • Figure 3: Root lattice $R_1$ of type $D_4$
  • Figure 4: Labels for positions in MOG (CS)
  • Figure 5: Hexadecagon and pentadecagon with two legs in $I_{1, 15}$ and $I_{1, 14}$
  • ...and 18 more figures

Theorems & Definitions (113)

  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 103 more